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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Rearranging (<a href="" class="xref" data-knowl="./knowl/eq1_7.html" title="Equation 1.4.1">(1.4.1)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_7.html">
\begin{equation*}
\frac{S(t+\Delta t)-S(t)}{\Delta t}=r S(t)+k.
\end{equation*}
</div>
<p class="continuation">The equation for continuous compounding of interest and continuous deposits is obtained by taking the limit <span class="process-math">\(\Delta t \rightarrow 0\text{.}\)</span> The resulting differential equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_7.html">
\begin{equation*}
\frac{\mathrm{d}\, S}{\mathrm{d}\, t}=r S+k
\end{equation*}
</div>
<p class="continuation">which can be solved with the initial condition <span class="process-math">\(S(0)=S_0\text{,}\)</span> where <span class="process-math">\(S_0\)</span> is the initial capital.</p>
<span class="incontext"><a href="sec_4-intro.html#p-13" class="internal">in-context</a></span>
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